President Donald Trump has invited Russian leader Vladimir Putin to visit America, in a move that drew startled laughter from a US intelligence chief.

"That's gonna be special!" said Director of National Intelligence Dan Coats, when he was told about the invitation during a live interview.

A row is continuing over Mr Trump's first summit with Mr Putin, in Helsinki, where they talked privately.

Opposition Democrats say there should be no more one-to-one talks.

"Until we know what happened at that two-hour meeting in Helsinki, the president should have no more one-on-one interactions with Putin," said the top Democratic senator, Chuck Schumer, in a statement. "In the United States, in Russia, or anywhere else."

Mr Trump's presidency has been clouded by allegations that Russian hackers meddled in the 2016 US presidential election in his favour. The Kremlin denies the allegations.

In Helsinki, Mr Putin offered access to 12 Russians indicted in absentia by the US authorities over the alleged interference, on condition the Russian authorities could question 12 Americans over a different case. Mr Trump first praised the suggestion as "incredible" but later rejected it.

Since his return from Finland, he or the White House have had to correct or clarify other comments regarding Russia, creating confusion and prompting the Democrats to demand details of his private talks with Mr Putin.

What do we know about Putin's potential visit?
Mr Putin, in power in Russia since 2000, last visited the US in 2015, when he met President Barack Obama, Mr Trump's predecessor, on the sidelines of the UN General Assembly in New York to discuss the conflicts in Ukraine and Syria.

On Thursday, White House press secretary Sarah Sanders tweeted that discussions about a visit by Mr Putin to Washington DC this autumn were already under way.

Russia's ambassador to Washington, Anatoly Antonov, said his country had always been open to the idea of a visit but it was "up to the Kremlin to decide how many summits are needed, and when".

The announcement appeared to come as a surprise to US intelligence chief Mr Coats, who was told about it during a live interview at the Aspen Security Forum in the state of Colorado.

He added that he did not yet know what Mr Trump and Mr Putin had discussed during their meeting, at which only the pair and their interpreters were present.

What was Putin's 'incredible offer'?
At the post-summit news conference in Helsinki, Mr Putin was asked whether he would extradite 12 Russian intelligence agents indicted in the US for hacking Democratic Party computers.

No extradition treaty exists between the two countries but Mr Putin said he would meet the US government "halfway".

He said that US investigators could question the 12 suspects inside Russia if, in turn, Russian investigators were allowed to question US citizens with regard to a case against financier Bill Browder.

Mr Browder was instrumental in the US imposing sanctions in 2012 on top Russian officials accused of corruption in the Magnitsky affair.

One of the Americans on Russia's list is a former US ambassador to Moscow, Michael McFaul.

The idea of allowing Russia to quiz US citizens sparked outrage and the US Senate voted 98-0 against it. Secretary of State Mike Pompeo said it was "not going to happen".

Mr McFaul tweeted his gratitude to the Senate.

At the news conference in Helsinki, Mr Trump said: "He [Mr Putin] offered to have the people working on the case come and work with their investigations with respect to the 12 people. I think that's an incredible offer."

Now, however, he says he "disagrees" with Mr Putin's proposal.

"I'm thankful that Donald Trump has no intention of handing me over to Vladimir Putin to have me killed in a Russian prison," Mr Browder told the BBC.

President Trump has also clarified remarks at the news conference in which he said he saw no reason for Russia to have meddled in the 2016 US election - despite US intelligence concluding just that.

Speaking to CBS News on Wednesday, Mr Trump said he held Mr Putin personally responsible for interfering in the election, and that he was "very strong on the fact that we can't have meddling".

Mr Putin has also described the summit as "successful" but warned "there are forces in the United States that are prepared to casually sacrifice Russian-US relations".